Latent Multivariate Maximal Lyapunov Exponents

Sensitive dependence on initial conditions is a defining quality of chaotic systems in which small differences in how two units begin may lead to large differences in these units later in time. This property has strong theoretical implications for interpretability and predictability of time based phenomena. However estimation of maximal Lyapunov exponents, a defining metric of sensitive dependence on initial condition, from data with measurement error is difficult. This impedes the study of sensitive dependence in psychological and behavior research as these areas of research tend to have relatively high levels of measurement error compared to other fields of study. This is unfortunate as many psychological phenomenon such as life satisfaction, drug use behaviors, and genetic differences in behavior are considered to display sensitive dependence on initial condition.
Many Psychological researchers use structural equation modeling (SEM) to account for measurement error when modeling behavioral data. SEM is a versatile method for modeling systems of linear equations capable of explicitly modeling measurement error. This dissertation seeks to use SEM as a means of estimating maximal Lyapunov exponents from data with measurement error. This estimated maximal Lyapunov exponent via SEM will be termed a Latent Lyapunov Exponent (LLE). First, a mathematical derivation of an SEM equivalent of the R-method for estimating maximal Lyapunov exponents is shown. Then a series of simulation studies compare the proposed method to currently established methods on bias, variance, and MSE. A separate simulation will then test the efficacy of the LLE method for a smaller number of samples. Extensions of LLE to multivariate space will then be discussed. Next, a real data example from socially anxious individuals over a number of weeks will serve as an illustration of the use of the LLE method. Finally, limitations, and future directions will be discussed.

I would like to thank the Jefferson Scholars Foundation for supporting my work.